Gimli is a 384-bit permutation which makes use of an internal 96-bit permutation which works on columns. Every 4 rounds starting from the 1st a "small swap" is performed and every 4 rounds starting from the 3rd round a "big swap" is performed, these work on the first 32-bits of each colomn (see https://gimli.cr.yp.to/linear.png)

The Chacha20 permutation is a 512-bit permutation with an internal 128-bit permutation which works on rows every odd round and diagonally every even round.

If we assume that the internal permutation of each algorithm is absolutely random, would the whole design be considered provably safe? What kind of safety margin does each design allow for? For example, would chacha20 with a random 128-bit internal permutation be just as secure as a 512-bit random permutation and would Gimli with a random 96-bit internal permutation be just as secure as a 384-bit random permutation?
If neither of the above designs is provably safe, is there something else that offers a provably secure way of constructing a bigger permutation out of a smaller one?

$\begingroup$Ohhh, so your question is about constructing a bigger public (unkeyed) permutation out of a small one in a provably secure way. In this case I'm not longer sure whether EME / Feistel are the right tools indeed (as they build bigger keyed PRPs out of small PRFs / PRPs)$\endgroup$
– SEJPM♦Mar 30 at 10:40